Video Transcript
Given that the area of the rectangle in the diagram is two 𝑥 cubed plus 𝑥 squared minus five 𝑥 minus three, find an expression for the width of the rectangle.
From the diagram, we see that the length of the rectangle is two 𝑥 plus three. We recall that the area of a rectangle is given by the product of its length and its width. This means we can determine the width of this rectangle by dividing its area by its length. In this case, we will be dividing a cubic polynomial by a linear polynomial. We can do this by using polynomial long division.
We begin the long division by dividing the leading terms of the dividend and the divisor. By dividing two 𝑥 cubed by two 𝑥, we get 𝑥 squared. We add this to the quotient. Next, we will need to subtract 𝑥 squared times the divisor from the dividend. We find that 𝑥 squared times two 𝑥 plus three is two 𝑥 cubed plus three 𝑥 squared. We carefully line up each term beneath its like term in the dividend. And we must subtract both terms, not just the first term. It may be helpful to first distribute the negative then combine like terms, as shown. So, we actually subtract both two 𝑥 cubed and three 𝑥 squared. The cubic terms cancel, and we are left with negative two 𝑥 squared minus five 𝑥 minus three. This is our new dividend.
As long as the degree of the new dividend is greater than or equal to the degree of the divisor, we will need to apply this process again, starting with the quotient of the leading terms. The quotient of negative two 𝑥 squared and two 𝑥 is negative 𝑥. So, we add negative 𝑥 to the quotient. Next, we multiply negative 𝑥 times the divisor, two 𝑥 plus three, to get the following expression: negative two 𝑥 squared minus three 𝑥. We then subtract this expression from the new dividend. To perform the subtraction, it may be helpful to first distribute the negative then combine like terms, as shown. The quadratic terms cancel, leaving us with negative two 𝑥 minus three.
This new dividend has the same degree as the divisor. So we should be able to complete one more round of division, starting with the quotient of negative two 𝑥 over two 𝑥. That equals negative one, so we add negative one to the quotient. Then, we multiply the divisor by negative one. We then subtract this product, negative two 𝑥 minus three, from the new dividend. This leaves us with a zero remainder. Therefore, the quotient of two 𝑥 cubed plus 𝑥 squared minus five 𝑥 minus three divided by two 𝑥 plus three is 𝑥 squared minus 𝑥 minus one. This is the expression that represents the width of the given rectangle.
